Non-Convergence Result for Conformal Approximation of Variational Problems Subject to a Convexity Constraint

Choné, Philippe; Meur, Hervé Le

doi:10.1081/nfa-100105306

Cited by 21 publications

(45 citation statements)

References 9 publications

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“…The first attempt in this direction, due to Kawohl and Schwab [13], consisted of using a first-order finite elements method. Unfortunately it turned out to be flawed: indeed, Choné and Le Meur [9] proved that, given a family of structured meshes M h , one can always find a convex function u that is not a limit of convex functions u h , with u h piecewise linear on M h . In other words, internal approximations with first-order finite elements are bound to fail, and the authors illustrate their point by numerical examples.…”

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confidence: 99%

“…We first apply our method to a benchmark problem, taken from Choné and Le Meur [9], where the solution can be computed explicitly, and we find that our method works in situations where they find they do not have convergence. We then estimate the minimizers for the well known Rochet-Choné problem (see [19]), where we find that our solution matches the theoretical one, to a problem in OTC pricing of securities [7], and to a risk-minimization problem as in [12].…”

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An algorithm for computing solutions of variational problems with global convexity constraints

Ekeland¹,

Moreno-Bromberg

2009

Numer. Math.

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We present an algorithm to approximate the solutions to variational problems where set of admissible functions consists of convex functions. The main motivation behind the numerical method is to compute solutions to Adverse Selection problems within a Principal-Agent framework. Problems such as product lines design, optimal taxation, structured derivatives design, etc. can be studied through the scope of these models. We develop a method to estimate their optimal pricing schedules. Mathematics Subject Classification (2000)49-04 · 49M25 · 49M37 · 65K10 · 91B30 · 91B32

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An algorithm for computing solutions of variational problems with global convexity constraints

Ekeland¹,

Moreno-Bromberg

2009

Numer. Math.

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“…In this case, imposing convexity becomes a local feature, and the number of linear constraints for convexity is proportional to the size of the mesh. Yet, Choné and Le Meur showed in [33] that we cannot approximate in this way all convex functions, but only those satisfying some extra constraints on their Hessian (for instance, those which also have a positive mixed second derivative ∂ 2 u/∂ x i ∂ x j ). Because of this difficulty, a different approach is needed.…”

Section: Numerical Methods From the Jko Schemementioning

confidence: 99%

{Euclidean, metric, and Wasserstein} gradient flows: an overview

2017

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This is an expository paper on the theory of gradient flows, and in particular of those PDEs which can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal transport). The starting point is the Euclidean theory, and then its generalization to metric spaces, according to the work of Ambrosio, Gigli and Savaré. Then comes an independent exposition of the Wasserstein theory, with a short introduction to the optimal transport tools that are needed and to the notion of geodesic convexity, followed by a precise description of the Jordan-Kinderlehrer-Otto scheme and a sketch of proof to obtain its convergence in the easiest cases. A discussion of which equations are gradient flows PDEs and of numerical methods based on these ideas is also provided. The paper ends with a new, theoretical, development, due to Ambrosio, Gigli, Savaré, Kuwada and Ohta: the study of the heat flow in metric measure spaces.

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“…On the other hand, handling the convexity constraint numerically in a consistent and efficient way is also a challenging problem which has received a lot of attention in the last fifteen years. Choné and Le Meur [15] have first identified specific difficulties, one of which being that one cannot use conformal convex finite-elements since they form, as the meshsize vanishes, a set of functions which is not (by far!) dense in the set of convex functions.…”

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confidence: 99%

An Iterated Projection Approach to Variational Problems Under Generalized Convexity Constraints

Carlier

Dupuis

2016

Appl Math Optim

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The principal-agent problem in economics leads to variational problems subject to global constraints of b-convexity on the admissible functions, capturing the so-called incentive-compatibility constraints. Typical examples are minimization problems subject to a convexity constraint. In a recent pathbreaking article, Figalli, Kim and McCann [19] identified conditions which ensure convexity of the principal-agent problem and thus raised hope on the development of numerical methods. We consider special instances of projections problems over b-convex functions and show how they can be solved numerically using Dykstra's iterated projection algorithm to handle the b-convexity constraint in the framework of [19]. Our method also turns out to be simple for convex envelope computations.

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Non-Convergence Result for Conformal Approximation of Variational Problems Subject to a Convexity Constraint

Cited by 21 publications

References 9 publications

An algorithm for computing solutions of variational problems with global convexity constraints

An algorithm for computing solutions of variational problems with global convexity constraints

{Euclidean, metric, and Wasserstein} gradient flows: an overview

An Iterated Projection Approach to Variational Problems Under Generalized Convexity Constraints

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